is the new extended incomplete beta function
. For m = 0, we must have p,q > 0 in (65) for convergence, and [sup.MC][B.sub.0,x](p,q) = [B.sub.x](p, q), where [B.sub.x](p,q) is the incomplete beta function
 defined as
In this work, we continue that line of investigation considering the incomplete beta function [B.sub.z](a, b).
The incomplete beta function [B.sub.z](a, b) reduces to the ordinary beta function B(a, b) when z = 1 and, except for positive integer values of b, has a branch cut discontinuity in the complex z-plane running from 1 to [infinity].
And conversely, by using repeatedly this formula, we have that [B.sub.z](a, b), with Ra [less than or equal to] 0, may be written as a linear combination of elementary functions of its three variables and an incomplete beta function with Ra > 0.
k > 0, a, b real, 0 [less than or equal to] u [less than or equal to] 1, where [beta](u,a + 1,b + 1) is the incomplete beta function
with parameters a and b.
Note that [.sub.z2][F.sub.1] (1, b; c; z) = [phi](b; c; z) is known as incomplete beta function
 Karl Pearson, ed., Tables of the Incomplete Beta Function
, Cambridge University Press, Cambridge, UK, first edition (1934).
where [B.sup.*] [[F.sup.1/[beta]] : [beta] + 1/[delta], 1 - 1/[delta]] is the incomplete beta function
with parameters ([beta] + 1/[delta], 1 - 1/[delta]), and B [[beta] + 1/[delta], 1 - 1/[delta]] is the beta function with parameters ([beta] + 1/[delta], 1 - 1/[delta]).
The area of a spherical cap can also be described using the incomplete Beta function. LEMMA 2.6.
where B (x; a, b) is the incomplete Beta function [6J and B (a, b) is the Beta function.
Keywords: Integral Inequalities; Hermite-Hadamard Inequality; -convex function; Beta and incomplete Beta Functions
; Holder's Inequality; Means for real numbers.
Among them are combinations of logarithms and exponentials; some trigonometric integrals; combinations of powers, exponentials, and logarithms; the incomplete beta functions
; and trigonometric forms of the beta function.